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One particle has a charge of [tex]-1.87 \times 10^{-9} \, C[/tex], while another particle has a charge of [tex]-1.10 \times 10^{-9} \, C[/tex]. If the two particles are separated by 0.05 m, what is the electromagnetic force between them?

The equation for Coulomb's law is [tex]F_e = \frac{k q_1 q_2}{r^2}[/tex], and the constant, [tex]k[/tex], equals [tex]9.00 \times 10^9 \, N \cdot m^2 / C^2[/tex].

A. [tex]-7.41 \times 10^{-6} \, N[/tex]
B. [tex]-3.70 \times 10^{-7} \, N[/tex]
C. [tex]7.41 \times 10^{-6} \, N[/tex]
D. [tex]3.70 \times 10^{-7} \, N[/tex]

Sagot :

GinnyAnswer
To find the electromagnetic force between the two particles, we'll use Coulomb's law, which is given by the formula:

[tex]\[ F_e = \frac{k q_1 q_2}{r^2} \][/tex]

where:
- [tex]\(k\)[/tex] is Coulomb's constant, [tex]\(9.00 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2\)[/tex],
- [tex]\(q_1\)[/tex] and [tex]\(q_2\)[/tex] are the charges of the particles,
- [tex]\(r\)[/tex] is the separation distance between the charges.

Given data:
- [tex]\( q_1 = -1.87 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( q_2 = -1.10 \times 10^{-9} \, \text{C} \)[/tex]
- [tex]\( r = 0.05 \, \text{m} \)[/tex]

Now substitute these values into the formula:

[tex]\[ F_e = \frac{(9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9})}{(0.05)^2} \][/tex]

First, calculate the numerator:

[tex]\[ (9.00 \times 10^9) \times (-1.87 \times 10^{-9}) \times (-1.10 \times 10^{-9}) \][/tex]

The product of the charges ([tex]\( q_1 \)[/tex] and [tex]\( q_2 \)[/tex]) will be positive because multiplying two negative numbers yields a positive result. Simplifying this, we get:

[tex]\[ (9.00 \times 10^9) \times (1.87 \times 10^{-9}) \times (1.10 \times 10^{-9}) = 9.00 \times 1.87 \times 1.10 \times 10^9 \times 10^{-9} \times 10^{-9} \][/tex]

[tex]\[ = 9.00 \times 1.87 \times 1.10 \times 10^{-9} \][/tex]

Next, calculate the denominator:

[tex]\[ (0.05)^2 = 0.0025 \][/tex]

Now substitute these into the expression for [tex]\( F_e \)[/tex]:

[tex]\[ F_e = \frac{9.00 \times 1.87 \times 1.10 \times 10^{-9}}{0.0025} \][/tex]

Calculate the values:

[tex]\[ 9.00 \times 1.87 = 16.83 \][/tex]

[tex]\[ 16.83 \times 1.10 = 18.513 \][/tex]

[tex]\[ F_e = \frac{18.513 \times 10^{-9}}{0.0025} \][/tex]

[tex]\[ F_e = 18.513 \times 10^{-9} \times \frac{1}{0.0025} \][/tex]

[tex]\[ F_e = 18.513 \times 10^{-9} \times 400 \][/tex]

[tex]\[ F_e = 7.4052 \times 10^{-6} \, \text{N} \][/tex]

Therefore, the electromagnetic force between the two particles is:

[tex]\[ \boxed{7.41 \times 10^{-6} \, \text{N}} \][/tex]

So the correct answer is:

C) [tex]\(7.41 \times 10^{-6} \, \text{N}\)[/tex]
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